We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.
|Title of host publication||Regulators IV|
|Subtitle of host publication||An international conference on arithmetic L-functions and differential geometric methods|
|Editors||A Chambert-Loir, J-H. Lu, M. Ruzhansky, Y. Tschinkel|
|Place of Publication||Birkhäuser|
|Number of pages||44|
|Publication status||Published - 2021|
|Name||Progress in Mathematics|